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# Using the cosine angle addition identity

CCSS.Math:

## Video transcript

we've got triangle a b c here it looks like a right triangle we can verify that it is because it satisfies the Pythagorean theorem 8 squared is 64 plus 15 squared is 225 64 plus 225 is 289 which is 17 squared so satisfies the Pythagorean theorem so we know we know that this right over here is a right triangle but what they're asking us is what is the cosign of angle ABC ABC which is this angle right over here plus 60 degrees now just like this I don't have any clear way of evaluating it but we do have some trig identities in our tool kit that we could use to express this in a way that we might be able to evaluate it in particular we know that the cosine of a plus B is equal to the cosine of a times the cosine of B minus the sine of a times the sine of B so we can do the exact same thing here with the cosine the cosine of angle ABC of angle a b c plus plus 60 degrees this is going to be equal to let me just write out the whole thing the cosine of angle ABC times the cosine of 60 degrees minus the sine of angle ABC times the sine of 60 degrees so this right over here is angle a BC this is angle a BC and this is 60 degrees and this is 60 degrees so let's evaluate each of these things what is what is the cosine of angle ABC going to be equal to I'll do that over here the cosine of angle a BC well we could go back to sohcahtoa let me write it down so Toa cosine is defined as the adjacent over the hypotenuse cosine is defined as the adjacent for this angle the adjacent over the hypotenuse so cosine of angle ABC is equal to 15 over 17 so this right over here is 15 over 17 what is the cosine of 60 degrees well for there we have to turn back to our knowledge of 30 60 90 triangles so if I have a triangle like this so let me do my best here construct a 30 60 90 right triangle so that's a 60-degree side this is the 30-degree side we know and we've seen this multiple times if the hypotenuse has length 1 the side opposite the 30-degree side is 1/2 and then the side opposite the 60-degree side is square root of 3 times this so it's square root of 3 over 2 so the cosine of 60 degrees if you're looking at this side right over here let me write it let me write it in a color I haven't used yet so I'm Kerry I care about the cosine of 60 degrees the cosine of 60 degrees is going to be equal to once again adjacent over hypotenuse 1/2 over 1 it's going to be equal to 1/2 over 1 which is equal to 1/2 now let's think about the sine now let's think about the sine of angle ABC ABC so that's this one right over here so we've already drawn we have our triangle right here sine is opposite over hypotenuse so opposite is has length 8 over hypotenuse is 17 so it's equal to 8 over 17 and then finally we need to figure out we need to think about what the sine of 60 degrees is well 4 at the 60 degree side of this right triangle opposite over hypotenuse so square root of 3 over 2 over 1 which is just square root of 3 over 2 so we have all the information we need to evaluate it this whole thing so this was the sine of 60 degrees this whole thing is going to evaluate to cosign of angle ABC is 15 over 17 times cosine of 60 degrees is 1/2 so times 1/2 and then we're going to subtract sine of ABC which is 8 over 17 and then times sine of 60 which is square root of 3 over 2 so times square root of 3 / - and now we just have to simplify a little bit so this is going to be equal to if I multiply 1/2 times this let's see that's going to be 15 over 30 415 over 34 - and let's see we're dividing by 2 so it's 417 I'll write this for square roots of 3 over 17 and that's about as simple as I could do it if I wanted I could sum I could have a common denominator here of 34 and and then I could add the 2 so it could be 8 square roots of 3 over 32 over 34 but that still won't simplify it that much so this is a reasonably this is a reasonable good answer for what they are asking for 15 over 30 4 - 4 square roots of 3 over 17